J4.1 PARAMETERIZATION OF INTERNAL WAVE BREAKINGDUE TO NEAR-INERTIAL SHEAR

JULIE C. VANDERHOFF ∗

Brigham Young University, Provo, Utah

ABSTRACT

Internal waves are continuously being generated and propagating through the ocean and atmosphere.Internal wave breaking can occur far from generation sites, and the resultant mixing can transportmomentum, heat, and pollutants across isopycnals, maintaining the energy balance in the ocean andpreventing a stagnant deep ocean. But global circulation models cannot resolve these motions andthey must be parameterized. For a completely accurate parameterization, all waves and their possibleensuing motions due to other waves, bottom topography, vortices, boundaries, etc. must be accountedfor in the computations. Although many interactions are possible as internal waves propagate, evidenceof constant large scale inertial motions in the ocean lead us to study the breaking of internal waveswhich propagate both aligned with and in opposition to large scale inertial waves. The results of thetwo types of interactions are dynamically different: one is a time-dependent critical level and the other acaustic interaction. These different types of interactions can lead to wave-breaking pre-or post-maturelydue to the time-dependence of the inertial waves. The interaction is modeled through integration of thefully nonlinear, inviscid, Boussinesq equations of motion. In general, breaking is found to occur withina particular region of the inertial wave, which shifts for small scale waves that approach the interactionwith different group velocities. Small-scale internal waves with the largest vertical wavelengths aremost likely to break immediately as they enter an inertial wave propagating in the opposite direction,where the smallest vertical scale waves are more likely to break in-between strong refraction sites, if atall. When propagating the same direction, the scale separation between the waves is also important indetermining breaking probability although in this case larger separation results in a higher probabilityof breaking. Wentzel-Kramers-Brillouin (WKB) ray tracing is used to supplement the fully nonlinearnumerical model. These statistics expand the reach of calculations from the simulations and comparewell with not only which waves are expected to break due to the time dependence, but also wherethey would be expected to break within the inertial wave, dependent on their properties. Results of themodels also compare well with observations from the Hawaiian Ocean Mixing Experiment (HOME).

1. Introduction

Internal waves are ubiquitous in the ocean and cancarry and dissipate energy throughout the ocean. In-teractions with other flows can lead to internal wavesteepening to a point of breaking, resulting in mix-ing of organisms, heat, and pollutants. This break-ing and mixing must occur to keep the deep oceansfrom being stagnant and to close the basic energy bal-ance. But locations of strong dissipation are not wellknown, nor are the specific mechanisms by which theyoccur. Vanderhoff, Nomura, Rottman, and Macaskill(2008) found small scale waves that propagate throughlarger scale inertial frequency waves with an initially

∗Corresponding author address: Julie C. Vanderhoff, Mechani-cal Engineering Department, Brigham Young University, 435 CTB,Provo, UT, 84602-4201. (email: jvanderhoff@byu.edu).

slow upward vertical group speed (slower than the up-ward phase speed of the long wave) and large verticalwavenumber have regions of greatly increased waveaction during strong refraction. Breaking regions werenot parameterized as a part of the study, though.

It is conjectured that mixing does not occur uni-formly over the entire ocean, which has been supportedby recent measurements which show an increase inmixing over topography (Polzin, Toole, Ledwell, andSchmitt (1997)). The Hawaiian Ocean-Mixing Ex-periment, HOME, (Pinkel, Munk, Worcester, Cor-nuelle, Rudnick, Sherman, Filloux, Dushaw, Howe,Sanford, Lee, Kunze, Gregg, Miller, Moum, Caldwell,Levine, an G. D. Egbert, Merrifield, Luther, Firing,Brainard, Flament, and Chave (2000), Pinkel and Rud-nick (2006)) was conducted to examine the processesthat lead to ocean mixing at a site of strong baro-

1

clinic tidal generation (Noble, Cacchione, and Schwab(1988), Holloway and Merrifield (1999)). HOME hadboth observational and computational components.Through satellite altimetry, HOME researchers Zaronand Egbert (2006) found that 26 gigawatts of tidal en-ergy is dissipated in the region of the Hawaiian Ridge.Some fraction of this radiates to the deep sea as lowmode baroclinic waves while the remainder is dissi-pated locally (Rainville and Pinkel (2006a,b), Mer-rifield and Holloway (2002)). HOME observationsdemonstrated intense baroclinic wave generation, withhigh levels of turbulence in the lower 1500 m of thewater column. Turbulent mixing rates decayed off-shore, and by 60 km away the rates had fallen to typi-cal open-ocean values. Documenting the cascade pro-cess, by which barotropic tidal energy is transferredacross a range of scales to eventual turbulent mix-ing, is a principal goal of HOME (Klymak, Moum,Nash, Kunze, Girton, Carter, Lee, Sanford, and Gregg(2006), Lee, Kunze, Sanford, Nash, Merrifield, andHolloway (2006), Klymak and Moum (2007a,b)).

The interaction of long and short internal wavesplays a role in this process. Even though baroclinictidal motions were most energetic at the Nearfieldsite, the shear is primarily associated with near iner-tial waves. Whether these waves are generated by thelocal wind, topographic interaction, or by non-linearinteractions with the baroclinic tide is a subject of cur-rent research. Here, the HOME observations are usedto set the scales of the inertial motions used in ray trac-ing and fully non-linear simulations of shorter wavespropagating through longer inertial waves. The shortwaves are assumed to be pre-existing, with low initialenergy, and their steepness is calculated as they propa-gate to asses locations of possible wave breaking.

The next section will cover the setup of each of thestudied media: observations, numerical simulations,and ray tracing. In Section 3 results will be presented.Section 4 will draw conclusions about these results.

2. Setup

This section will cover the different setups of the ob-servations, ray tracing calculations and numerical sim-ulations.

a. Observational Setup

The HOME Nearfield experiment was conducted onthe Kaena Ridge, a submerged extension of the hawai-ian island of Oahu. The Ridge extends west-north-westfrom Oahu for about 60 km, half of the distance toKauaii. During September-October, 2002, the FLoat-ing Instrument Platform, FLIP, was moored as shown

FIG. 1. Site of the 2002 HOME Nearfield Experiment.The blue circle represents the location of the ResearchPlatform FLIP, the red circles are the locations of ancil-lary moorings. The solid lines denote cross-ridge andalong-ridge directions.

in Fig. 1, 21.7 North, 158.6 West, on the south-westedge of the Ridge crest. At the location of FLIP thecrest depth is about 1100 meters, with surrounding off-ridge areas at 5 km depth. Instruments deployed onFLIP, including an eight-beam, coded-pulse Dopplersonar that measured velocity from 50-800 m with 4mvertical resolution. Two CTDs (current-temperature-depth, Seabird SBE 911 ) profiled vertically from 20meters to 820 meters depth at 4 minute intervals. The3.5 m/s profiling speed leads to a resulting 1.1 metervertical resolution in temperature, salinity and poten-tial density. For further information of the setup of theexperiment see Klymak, Pinkel, and Rainville (2007).

Slopes as steep as 1:4 define the north-north-east andsouth-south-west sides of the ridge. The ridge is ori-ented roughly normal to local semi-diurnal barotropictidal flow. The S2 (12 hour semidiurnal solar) tidal cur-rent has amplitude 2.8 cm/s East and 5.2 cm/s North.The K1 (24 hour diurnal solar) tidal current has am-plitude 3.2 cm/s East and 4.6 cm/s North. The M2

(semidiurnal lunar - 12 hour 25 minute) tidal currenthas amplitude 6.4 cm/s East and 11.7 cm/s North, andis the dominant tide. It has a pronounced fortnightlycycle. Above 500 meters, energy and momentumfluxes are upward and southward (1dyne/cm2 duringspring tide. Below 500 meters the fluxes are upwardand northward. Above the ridge crest, power spectraof horizontal velocity and vertical displacement havepronounced D2 (semi-diurnal - 12 hour) peak. Thereis little evidence of a D2 peak in the shear, as shownlater. The cruise covered two fortnightly cycles. Thefirst neap tide was covered from year day 257 (Septem-

2

FIG. 2. Observations from sonar data over KaenaRidge for one week (year day 260 to year day 267)over all depths (100 meters to 800 meters). Inertial, di-urnal, and semidiurnal frequencies are labeled. A linewith slope of −2 is superimposed on the frequencygraph for the velocity. (a) Frequency spectrum forcross-ridge velocity averaged over all depths (371 datapoints). (b) Frequency spectrum for cross-ridge shear,Uz , averaged over all depths (371 data points).

ber 14, 2002) to year day 261 (September 18, 2002).The first spring tide was from year day 262 (Septem-ber 19, 2002) to year day 269 (September 26, 2002).

Cross-ridge velocity and shear frequency spectracalculated over one week, from year day 260 to yearday 267 for all depths are shown in Fig. 2. Peaks can beseen in the observations in both the velocity and shearfrequency spectra at the inertial, diurnal, and semidiur-nal frequencies. The strongest peak in the velocity cor-responds to the tidal frequency, yet the strongest peakin the shear spectrum corresponds to the local inertialfrequency - suggesting a strong inertial wave presence.An approximately −2 high frequency slope and highvertical wavenumber slope can also be seen.

Preliminary observational data taken on FLIPpresent a strong argument for a need to understandhow the squared strainrate field (∂(∂ζ/∂t)/∂z =∂2ζ/∂z∂t), which is a measure of high frequency

wave activity, is affected by the near-inertial waves.This can be seen in Fig. 3 a week-long record of thecross-ridge shear normalized by N and the strainratesquared, which represents high frequency wave activ-ity. The sideways chevrons in the shear are characteris-tic of upward and downward propagating near-inertialwaves, with periods of approximately 24 to 30 hours,frequency of about f to 1.3f . Later, wave breakingregions with respect to these regions will be discussed.

b. The idealized problem

In the ray tracing and numerical simulations we con-sider the case of a packet of short waves approach-ing a single inertia packet either from above or be-low, as described in Vanderhoff, Nomura, Rottman,and Macaskill (2008), where a steady shear may bepresent as well. The coordinate system is (x, y, z) withz positive downward, x positive northward, and y posi-tive eastward. We assume that the buoyancy frequencyN and the Coriolis parameter f are both constant.

The inertial packet has wavenumber K = (0, 0,M),whereM = 2π/λi and λi is the vertical wavelength ofthe wave. The corresponding velocity field is uniform,horizontally, u = (u, v, 0), but confined in the verticalby a Gaussian envelope:

u+ iv = u0 e−z2/2L2

ei(Mz−ft) (1)

where L and u0 are constants, real and complex re-spectively. The envelope of the inertia-wave packetassumed stationary, since the vertical component ofthe group velocity vanishes at the inertial frequency.The phases move vertically through the packet at speedc = f/M , assumed positive to match the observa-tions analyzed. The short waves have wavenumberk = (k, 0,m), with k constant, and intrinsic frequencyω, which is the Doppler-shifted frequency, where

ω2 = (N2k2 + f2m2)/(k2 +m2) . (2)

The vertical group velocity cg = ∂ω/∂m is negative ifm is positive and positive if m is negative.

The vertical displacement of the short waves is ζ =ζ0 exp(iθ), from which the wavenumber and wave fre-quency are given by k = ∇θ and ω = −θt, respec-tively, and where ω = ω + ku. The wave-energy den-sity E is related to ζ0 by

E =12ρ0ζ

20N

2

[1 +

(fm

Nk

)2]

(3)

where ρ0 is the mean density of the fluid.The numerical simulations are initialized at time

t = 0 with a short-wave packet whose vertical dis-placement field ζ(x, z, t) has the initial form

3

FIG. 3. (a) Cross-ridge shear normalized by buoyancy frequency, and presented in a reference frame moving ver-tically with isopycnal surfaces. (b) Strain rate squared [1/s2] calculated from the profiling CTDs is also presentedin this isopycnal following frame. The dark blue regions (circled) are relatively devoid of high-frequency waves.

ζ(x, z, 0) = Reζ0 e−(z−z0)

2/2`2ei(kx+mz)

(4)

where ` and z0 are real constants and ζ0 is a complexconstant. The initial vertical position z0 is specifiedsuch that the short-wave packet is above the inertia-wave packet if cg < 0, and below it if cg > 0. Thevertical derivative of the vertical displacement field isthe wave steepness, derived from the dispersion rela-tion and (3),

ζz = −m

∣∣∣∣∣(

2Aωρ0

)1/2

N−1

∣∣∣∣∣ . (5)

When the wave steepness is greater than unity the shortwaves are expected to break.

For the ray tracing and numerical simulation resultsshown in this paper, we use the following ocean pa-rameters, which are defined by the observations: M =2π/(100 m), k = M/2, f = 10−4 s−1, N/f = 75,and u0 = 0.05 m/s. For the numerical simulations,the initial steepness |ζz| = |mζ0| = 0.1, where sub-script z represents the partial derivative with respect toz, ML =

√2π/10, and `/L = 0.75. We will alter

the vertical wavenumber, m, to realize different groupspeeds of the short wave.

c. Ray Theory

Using ray theory we can calculate approximately thebehavior of the short wave encounter with the inertialwave group. To do this we assume that the inertialwave is both unaffected by the short wave interactionand has a much larger length scale than that of the shortwave. Also we assume the short wave is determined bythe linear dispersion relation. Then an evolution equa-tion in characteristic form can be found for k. For fur-ther detail see Vanderhoff et al. (2008).

1) THE RAY EQUATIONS

The ray-tracing results in this paper are obtainedwith the following pair of ray equations, for the verticalposition of the ray path and the vertical wavenumberrespectively:

dz

dt= cg,

dm

dt= −k∂u

∂z. (6)

Here d/dt = ∂/∂t + cg∂/∂z. Because the expres-sion (1) has no dependence on x or y, the horizon-tal components (k, 0) of the wavenumber of the shortwaves are conserved along the ray.

In a reference frame moving at the inertial-wavephase speed c, the inertial current appears steady. So-lutions then exist for which the short-wave frequencyin the inertial-wave reference frame

4

Ω = ω + ku− cm ≈ constant. (7)

The trapped solutions have the short-wave grouppermanently confined within one wavelength of theinertia-wave, so that there are regions of the inertia-wave train where the short-wave group cannot propa-gate. The boundaries of these regions are called caus-tics and are the curves in (t, z) where neighboring rayscross (see Vanderhoff et al. (2008) Fig. 2). For ouridealized model, caustics occur when

cgz = Cz , (8)

where the capitals represent the inertial wave. Criticallevels occur when the relative frequency of the shortwave goes to zero.

d. Numerical Simulations

Numerical results are obtained by integrating thefully nonlinear inviscid, Boussinesq equations of mo-tion. In their vorticity-streamfunction form, these are:

∂2ψ

∂x2+∂2ψ

∂z2= q (9)

∂q

∂t− J(ψ, q)− ∂σ

∂x− f ∂v

∂z= 0 (10)

∂v

∂t− J(ψ, v) + fu = 0 (11)

∂σ

∂t− J(ψ, σ)−N2w = 0, (12)

where q is the y-component of vorticity and J(ψ, q)the Jacobian with respect to (x, z). Here the fluid ve-locity u = (u, v, w), and the stream function ψ isdefined such that u = ∂ψ/∂z, w = −∂ψ/∂x, andq = ∂u/∂z − ∂w/∂x. The scaled density perturba-tion due to the presence of internal wave motions isσ = gρ′/ρ0 where g is the acceleration due to gravity;the density ρ = ρ′ + ρ0, with ρ0(z) the mean densityprofile. Because of rotation, there is a nonzero v field,but all variables are assumed to be independent of y.

Periodic boundary conditions are imposed in boththe x- and z-directions, and the equations are solvedusing a Fourier spectral collocation technique withRunge-Kutta time stepping. The computational do-main contains one horizontal wavelength of the shortwaves in the horizontal direction and one or more ver-tical wavelength(s) of the inertia waves in the verticaldirection. There are 512 grid points in the vertical di-rection, but only 16 grid points in the horizontal di-rection. The low horizontal resolution suffices for thewave propagation to an increased steepness, but doesnot resolve any breaking.

3. Results

Wave-breaking is defined when isopycnals are ver-tical, ζz > 1, leading to overturning within the fluidand resulting turbulence. This can be calculated inthe numerical simulations and observations by finding∆ζ/∆z. For calculating wave steepness in ray theoryequation (5) is used.

a. Wave Breaking in Observations

The observational results of calculating the break-ing parameter, ζz , from the CTD data over two daysand 200 meters depth are shown in Fig. 4b. The cor-responding filtered inertial shear is shown in Fig. 4a.These results show a strong relationship betweenbreaking and negative shear. Some of the strongestwave breaking regions are highlighted in Fig. 4. Butwhat does the wavefield look like in these regions?

When low-frequency waves are filtered out, the av-erage Reynolds stress (RS),uw, over two days is shownby the solid lines in Fig. 5. Upward propagating wavesare on the left, and downward on the right. For upwardpropagating waves positive RS corresponds to wavepropagation in negative-x, and for downward propaga-tion, positive-x. The dashed lines are the average shearduring breaking (x5x103) over the two days. Both RSare near the same order, but each has a preferred di-rection during different sign shear breaking regions. Inthe upper region, where some of the strongest break-ing regions occur, the downward waves have positiveRS, corresponding to positive-x traveling waves. Thisdirectionality will become important while discussingthe possible interactions leading to breaking.

Fig. 6 is an average of the RS over time at each depthmultiplied by the shear and shown only in breakingregions. The data has been filtered to include onlyhigh frequency waves and Fig. 6a is upward propa-gating waves and Fig. 6b is downward propagatingwaves. Here, these are upward propagating waves andthose propagating in the positive (northward) directionhave a negative RS and are expected to break in neg-ative shear regions (inertial wave horizontal velocityincreasing as the short waves propagates upward), aswell known critical layer theory shows. Those propa-gating southward have a positive Reynolds stress andare expected to break in positive shear regions. Thus,the product of the shear and RS should be positive inbreaking regions. Regions in Fig. 6a where the prod-uct is negative are not explained by upward propagat-ing waves interacting with the inertial wave. Thus thebreaking mechanism should be due to downward prop-agating waves.

The downward propagating waves visualized inFig. 6b includes both positive and negative product

5

FIG. 4. Observational analysis over Kaena Ridge for two days over 200 meters depth. Circled regions of strongestbreaking, with a line down another region of strong breaking. (a) Inertial shear divided by buoyancy frequency.(b) Wave breaking map calculated from CTD data. The colorbar represents ζz .

FIG. 5. Observational analysis over Kaena Ridge for two days over 200 meters depth. Reynolds stress, uw,[m2/s2] (solid line) and shear [1/s] (x5x103) where breaking is occurring (dashed line) are each averaged overtwo days. (a) Filtered for upward propagating, high frequency waves only. (b) Filtered for downward propagating,high frequency waves only.

6

FIG. 6. Observational analysis over Kaena Ridge for two days over 200 meters depth. Reynolds stress (uw)is averaged over two days and multiplied by the shear at each point. Results are only shown where breaking isoccurring. (a) Filtered for upward propagating, high frequency waves only. (b) Filtered for downward propagating,high frequency waves only. The colorbar represents a positive (white) or negative (black) product.

regions where the product was negative in Fig. 6a(where the upward traveling waves were not break-ing). Specifically, the strong breaking region, circledin Fig. 4b, and the early time region near 460m depth,the product is negative. Fig. 5b shows in that spatialregion, The RS is positive, meaning small-scale down-ward propagating waves are dominated by northwardpropagation. Although other interactions may also beoccurring, this region was originally singled out dueto the strong inertial wave presence, and thus the mainbreaking would be expected to be due to interactionsbetween small-scale waves present and the large scaleinertial wave. The goal of the next two subsections isto explain breaking in each region due to small-scale,high frequency wave interactions with an inertial wave.

Two types of interactions will be analyzed in an ef-fort to understand the dynamics of the wave interac-tions in the observations. High-frequency, small-scaleinternal waves will interact with a large-scale iner-tial frequency wave with downward propagating phasespeed, as is seen over depths of 400 meters to about600 meters as discussed above. The small-scale waveswill approach the inertial wave from above, below, andpropagating in both positive- and negative-x directions.This should account for the main expected types of in-teractions in this region.

b. Wave Breaking in Ray Tracing

In an effort to explain the breaking phenomenonseen in the observations, ray tracing of small-scale in-ternal waves are set to interact with and an inertialwave with downward propagating phases.

Fig. 7 shows the ray lines, where locations of strongrefraction are outlined by filled in ellipses, and thecorresponding ζz values for a fast short wave. Thesevalues are estimated at the location of strong refrac-tion, caustic, with the corrected amplitude. In ray trac-ing calculations the amplitude of the short wave ap-proaches infinity as the caustic is approached and anAiry function relationship is used to estimate the max-imum amplitude in this region. Locations of positiveand negative shear which border the ellipses is shownin Fig. 7, where the upper left portion of the ellipse cor-responds to positive background shear and the lowerright portions correspond to negative background shearwhen the rays have a positive horizontal wavenumber,k. This is opposite for waves traveling in the nega-tive x-direction (which have negative k values). In thelower portion of Fig. 7 there is a large increase in wave-steepness at the caustic, the steepness increases by over15 times the original. Also, while traveling inbetewenthe phases the steepness reaches almost 15 times itsoriginal value. This first occurs in a region of posi-tive shear, and thus breaking would first be expectedin a region of positive shear. This results in the signof the product of the Reynolds stress and the shear aspositive, for both northward and southward, downwardtraveling short waves through an upward traveling in-ertial wave.

This will not be true for the downward propagatingslower traveling waves in Fig. 8, which do not have thesame first refraction seen in Fig. 7, but their steepnesswill begin to increase between the phases of the iner-tial wave. These waves may break after the first strong

7

FIG. 7. Ray paths for a downward propagating short wave with a faster vertical group speed than the verticalphase speed of the inertial wave, m/k = 3. The waves are propagating in the positive-x direction, resulting ina positive Reynolds stress. The filled in ellipses are outlined by locations of strong refraction. The positive andnegative signs depict the sign of the background shear at each location; positive above the strong refraction region,and negative below. Corresponding ζz/ζz0 values along the ray are plotted below. If the initial steepness = 0.1,then the steepness is greater than 1 when the ratio of steepness to initial steepness = 10, where the horizontal lineis drawn. The interaction is symmetric about the zero shear line (vertical line), but since the short wave reachesthe max steepness in the region of positive shear first it may lose its energy in the positive shear region, leavingless energy and a smaller probability of breaking in the negative shear region. This is the only asymmetry in theshort wave interactions, where breaking would be initiated. Negative-x traveling waves will have the oppositeasymmetry.

FIG. 8. As in Fig. 7, but for a short wave with a slower vertical group speed than the vertical phase speed of theinertial wave, m/k = 35.

8

refraction, in a positive shear region, or during the sec-ond in a negative shear region. Thus the product of theReynolds stress and background shear during break-ing may be either positive or negative. These, slower,small scale waves reaching caustics may explain theobservations showing breaking when the sign of theReynolds stress shear product is negative. These aresome of the strongest breaking regions in the observa-tions.

Waves propagating upward will not reach caus-tics, but will approach critical levels as they propa-gate. Steady critical levels have well known properties.Here, upward, northward traveling waves with a neg-ative RS will approach a critical level and most likelybreak in regions of negative shear. Southward travelingwaves have the opposite result, and therefore the samefinal product sign, positive. Thus, upward propagatingwaves are not expected to explain the regions in theobservations which show a negative RS, shear product.

Short waves, approaching an inertial wave propagat-ing in the same vertical direction, will most likely be-gin to break in a region where the horizontal velocityof the background increases in the direction of the hor-izontal short wave group speed as the short wave prop-agates vertically (critical level interaction). Thus if thez-direction is positive downwards, and short waves arepropagating in the positive x-direction, as the back-ground velocity becomes more positive as the shortwaves propagate in negative-z the short waves maybreak. This is a region of negative shear for a nega-tive RS value. Fig. 9 displays the shear values when thebreaking threshold is reached for 400 rays for fast (left)and slow (right) short waves. Short wave steepness iscalculated and if it is greater than the threshold (steep-ness = 1) we assume it breaks and cut the amplitudedown to 80% of the maximum for breaking. We let thewaves propagate into the breaking region and cut themoff at their maximum steepness. Then we kept this per-cent loss and calculated the total lost over the life of thewave. Sometimes it would break more than once. Notethe product of the shear and Reynolds stress will bepositive for all except case Fig. 9d. As expected fromthe theory and discussion of Fig. 5, slowly travelingdownward, northward propagating waves with causticinteractions can explain the strong breaking regions inthe observations.

c. Wave Breaking in Numerical Simulations

A wave breaking map for the numerical simulationof a small-scale, fast propagating wave approachingan inertial wave from below is shown in Fig. 10b,where the initial wave steepness is ζz = 0.8. Nextto it, Fig. 10a, is the corresponding background wave

shear field. In this setup, where the short waves aretraveling upward in the positive x-direction (negativeReynolds stress), breaking will be in regions wherethe background shear is decreasing with increasingdepth (negative shear): the region within the back-ground phase where a critical level begins to be ap-proached. These waves are the same type as in Fig. 9a.Slowly traveling waves (as in Fig. 9b) have the sameresult, although the approach to the critical level occurssooner. Testing a few different waves, allowing themaximum background velocity and the initial steep-ness to change, about 70-90% of breaking for occursin the expected negative shear, resulting in a positiveshear, RS product. Since short-waves traveling upwardin the negative-x direction (positive RS) break in posi-tive shear regions, they also have the most breaking inregions of a positive shear, RS product. They are notshown here because the propagation dynamics are thesame.

Fig. 11 shows the interaction when fast travelingdownward propagating short waves interact with theinertial wave. Completely different dynamics are oc-curring where strong refraction dominates the interac-tion instead of critical levels (Vanderhoff et al. (2008),Sartelet (2003a,b)). An estimate of breaking from thisinteraction shows 90% of the breaking is occurringin regions of positive shear. This again results in apositive shear, RS product, and can describe some ofthe breaking regions in Fig. 6b with positive productswhich corresponded to negative products in Fig. 6a (re-gions where upward propagating waves most likely donot support breaking). Again, signs are opposite whenshort waves are propagating in the negative-x direction,thus the dynamics and final product sign are the same.Although this explains more of the breaking regionswithin the inertial wave, breaking occurring during thenegative sign product of shear and RS has not yet beendescribed.

The final possibility for breaking in these regions(assuming the main interactions here are betweensmall-scale high-frequency waves and a large scale in-ertial wave) is the slowly traveling downward propa-gating waves of Fig. 9d and Fig. 5. Numerical simula-tions of this interaction are shown in Fig. 12. Breakinghere occurs partially in regions of positive shear, andpartially in regions of negative shear. Initially breakingis in regions of positive shear, but as the wave contin-ues through the interaction it begins to break in neg-ative shear regions as well. This interaction can de-scribe the final breaking regions within the observa-tions, where the product of the shear and RS is neg-ative. In the region of strong breaking, RS was posi-tive from Fig. 5 (downward, positive-x traveling as in

9

FIG. 9. Small-scale waves propagating through an inertial wave with phases propagating downward as in the oceanregion. Shear values when breaking threshold is reached (open circle), at the max of the breaking region (cross),and at the final location of breaking (asterisk), for different values of initial steepness, (Ak/ω)1/2. After breakingthe short wave has 80% of the energy it had when it reached the breaking threshold. If the shear is positive itis assigned a value of +1 and if negative it is -1. These values are averaged over 400 rays started at differentinitial slopes and depths. All small-scale waves are propagating in the positive-x direction. (a) Small-scale wavespropagating upward with fast vertical group speed. Reynolds stress is negative. (b) Small-scale waves propagatingupward with slow vertical group speed. Reynolds stress is negative. (c) Small-scale waves propagating downwardwith fast vertical group speed. Reynolds stress is positive. (d) Small-scale waves propagating downward with slowvertical group speed. RS is positive. Note the product of the shear and Reynolds stress will be positive for allexcept case (d)

10

FIG. 10. Numerical simulation of a small-scale wave approaching an inertial wave from below. The vertical groupspeed of the small-scale wave is much faster than the downward vertical phase speed of the inertial wave, initiallym/k = −3 (negative RS), and ζz = 0.8. (a) Background shear [1/days]. (b) Possible breaking map. The colorbarrepresents ζz . Notice most of the breaking occurs in regions of negative shear.

FIG. 11. As in Fig. 10, but short wave approaching from above, RS positive.

11

FIG. 12. As in Fig. 11, but short wave approaching slowly from above, RS positive.

Fig. 12) and shear was negative, resulting in a neg-ative product and the same type of waves seen here.Fig. 2 also shows the frequency spectrum, where thereis more wave energy at lower, high-frequencies, whichare these slower traveling waves.

4. Discussion

The results shown here have given us insight intoone of the possible mechanisms of short wave break-ing in the ocean. This is when small-scale, high-frequency internal waves propagate upward and down-ward through the ocean and interact with constantlypresent large-scale inertial waves. Observations haveshown these phenomenon occurring, and this analysisof small-scale, large-scale wave-wave interactions candescribe the breaking phenomenon seen. Strong break-ing was found in regions of negative shear. Small-scalewaves propagating in all directions were present, withthe lowest of the high-frequency waves being promi-nent. Locations where upward traveling waves can-not explain the shear at breaking locations, downwardtraveling waves are dominated by northward propagat-ing waves which explain breaking in negative shear re-gions due to strong refraction zones only found withina propagating inertial wave. Other locations of break-ing unexplainable by a conventional critical level anal-ysis can be explained by downward propagating wavesin the southward direction strongly refracting in neg-ative shear locations. Ray tracing supports the analy-sis through a statistical analysis of many representativewave-wave interactions.

These types of wave-wave interactions may oc-

cur anywhere in the ocean where small-scale internalwaves and inertial waves are present. Although in theregion studied here, just over topography, there is muchhigh-frequency internal wave activity due to the tidalflow over the topography, internal waves can be presentin any area where the ocean is stratified (all but the up-per mixed layer mainly). Inertial waves are large scaleand have been observed as a regular phenomeon. Thusbreaking due to these interactions may also provide in-sight into mixing occurring in the deep ocean and farfrom strong internal wave generation sites.

More analysis of other specific observational regionsis suggested for a more full understanding of the pro-cesses dominating the breaking. These results werecalculated with two-dimensional simulations and the-ory, yet three-dimensional simulations would be moreaccurate as the short waves begin to become unstableand are necessary to quantify wave breaking. It is alsonoted that the argument put forth here is not statingthese are the only types of interactions occurring, butthat they are highly probable and comparable.

REFERENCES

Holloway, P. E., and Merrifield, M. E., 1999: Internal tidegeneration by seamounts, ridges, and islands. J. Geo-phys. Res., 104, 25,937–25,951.

Klymak, J. M., and Moum, J. N., 2007a: Oceanic isopy-cnal slope spectra: Part I - Internal waves. J. Phys.Oceanogr., 37, 1215–1231.

Klymak, J. M., and Moum, J. N., 2007b: Oceanic isopycnalslope spectra: Part II - Turbulence. J. Phys. Oceanogr.,37, 1232–1245.

12

Klymak, J. M., Moum, J. N., Nash, J. D., Kunze, E., Girton,J. B., Carter, G. S., Lee, C. M., Sanford, T. B., andGregg, M. C., 2006: An estimate of tidal energy lost toturbulence at the Hawaiian ridge. J. Phys. Oceanogr.,36, 1148–1164.

Klymak, J. M., Pinkel, R., and Rainville, L., 2007: Directbreaking of the internal tide near topography: KaenaRidge,Hawaii. J. Phys. Oceanogr., in press.

Lee, C. M., Kunze, E., Sanford, T. B., Nash, J. D., Merrifield,M. A., and Holloway, P. E., 2006: Internal tides andturbulence along the 3000-m isobath of the HawaiianRidge. J. Phys. Oceanogr., 36, 1165–1183.

Merrifield, M. A., and Holloway, P. E., 2002: Model esti-mates of m2 internal tide energetics at the HawaiianRidge. J. Geophys. Res., 107, 3179–3191.

Noble, M., Cacchione, D. A., and Schwab, W. C., 1988:Observations of strong mid-pacific tides above HorizonGuyot. J. Phys. Oceanogr., 18, 1300–1306.

Pinkel, R., Munk, W., Worcester, P., Cornuelle, B. D., Rud-nick, D., Sherman, J., Filloux, J. H., Dushaw, B. D.,Howe, B. M., Sanford, T. B., Lee, C. M., Kunze, E.,Gregg, M. C., Miller, J. B., Moum, J. M., Caldwell,D. R., Levine, M. D., an G. D. Egbert, T. B., Merrifield,M. A., Luther, D. S., Firing, E., Brainard, R., Flament,P. J., and Chave, A. D., 2000: Ocean mixing studiednear Hawaiian Ridge. Eos, Trans. Am. Geophys. U., 81,545–553.

Pinkel, R., and Rudnick, D., 2006: Introduction to a series ofpapers from the Hawaiian Ocean Mixing Experiment.J. Phys. Oceanogr., 36, 965–966.

Polzin, K. L., Toole, J. M., Ledwell, J. R., and Schmitt,R. W., 1997: Spatial variability of turbulent mixing inthe abyssal ocean. Science, 276, 93–96.

Rainville, L., and Pinkel, R., 2006a: Baroclinic energy flux atthe Hawaiian Ridge: Observations from the R/P FLIP.J. Phys. Oceanogr., 36, 1104–1122.

Rainville, L., and Pinkel, R., 2006b: Propagation of low-mode internal waves through the ocean. J. Phys.Oceanogr., 36, 1202–1219.

Sartelet, K. N., 2003a: Wave propagation inside an inertiawave. Part I: Role of time dependence and scale sepa-ration. J. Atmos. Sci., 60, 1433–1447.

Sartelet, K. N., 2003b: Wave propagation inside an inertiawave. Part II: Wave breaking. J. Atmos. Sci., 60, 1448–1455.

Vanderhoff, J. C., Nomura, K. K., Rottman, J. W., andMacaskill, C., 2008: Doppler spreading of internalgravity waves by an inertia-wave packet. J. Geophys.Res., 113, C05018, doi:10.1029/2007JC004390.

Zaron, E. D., and Egbert, G. D., 2006: Estimating open-ocean barotropic tidal dissipation: The hawaiian ridge.J. Phys. Oceanogr., 36, 1019–1035.

13